UDC 519.21

MATHEMATICS

I. A. IBRAGIMOV

ON COMPLETE REGULARITY OF MULTIDIMENSIONAL STATIONARY PROCESSES

(Presented by Academician Yu. V. Linnik on 18 IV 1968)

The purpose of the present note is to obtain analogues of Theorem 1 from \((^1)\) and Theorem 4 from \((^2)\) for the case of multidimensional completely regular stationary processes (cf. \((^3,\,^4)\)).

1. Let first \(\xi(t) = (\xi_1(t), \ldots, \xi_n(t))\) be a stationary process with discrete time \(t = \ldots, 1, 0, 1, \ldots\). The process \(\xi(t)\) is completely regular if

\[ \sup |\mathbf{E}x\bar y| = \rho(\tau)\longrightarrow 0,\qquad \tau\to\infty, \]

where the supremum is taken over all \(x \in H_{-\infty}^{0}\), \(y \in H_{\tau}^{\infty}\), \(\mathbf{E}|x|^2=\mathbf{E}|y|^2=1\), and \(H_a^b\) denotes the linear closed (in the mean-square sense) span of the random variables \(\xi_j(t)\), \(j=1,\ldots,n\), \(a\le t\le b\). Denote by \(f(\lambda)=\|f_{ij}(\lambda)\|\) the spectral matrix density of \(\xi(t)\).

Theorem 1. The spectral density \(f(\lambda)\) of a completely regular process is always representable in the form

\[ f(\lambda)=\mathbf{P}(e^{i\lambda})\,g(\lambda)\mathbf{P}^*(e^{i\lambda}),\qquad \mathbf{P}=\|P_{ij}\|,\qquad g=\|g_{ij}\|. \]

Here \(\mathbf{P}(z)\) is a diagonal polynomial matrix, and the primitives \(G_{ij}(\lambda)\) of the functions \(g_{ij}(\lambda)\) satisfy the condition

\[ \omega_{ij}(\delta)= \sup_{\lambda,\,|h|\le\delta} \frac{|G_{ij}(\lambda+h)+G_{ij}(\lambda-h)-2G_{ij}(\lambda)|} {|G_{ii}(\lambda+h)-G_{ii}(\lambda)|^{1/2}|G_{jj}(\lambda+h)-G_{jj}(\lambda)|^{1/2}} \longrightarrow 0. \tag{1} \]

We give the scheme of the proof of this theorem. By Theorem 1 from \((^1)\), for all \(i=1,\ldots,n\),

\[ f_{ii}(\lambda)=|P_{ii}(e^{i\lambda})|^2 g_{ii}(\lambda), \]

where \(P_{ii}(z)\) is a polynomial with roots on \(|z|=1\), and \(G_{ii}(\lambda)\) satisfies condition (1). We now define the matrix \(\mathbf{P}(z)\) as the diagonal matrix with diagonal elements \(P_{ii}(z)\), and then define the matrix \(g(\lambda)\) from the equality

\[ f(\lambda)=\mathbf{P}(e^{i\lambda})g(\lambda)\mathbf{P}^*(e^{i\lambda}). \]

Lemma 1. The matrix \(g(\lambda)\) is the spectral matrix density of a stationary completely regular process.

In proving Lemma 1 it suffices to consider only the case when one of the diagonal elements of the matrix \(\mathbf{P}(e^{i\lambda})\) has the form \(e^{i\lambda}-e^{i\theta}\), while all the others are equal to one. In this case the proof is carried out in the same way as the proof of Lemma 3.1 from \((^1)\).

Put

\[ h_i(t;\mu)=h_i(t)=\frac{1}{t}\int_{-\pi}^{\pi} \frac{\sin^2(t\lambda/2)}{\sin^2(\lambda/2)}\,g_{ii}(\lambda+\mu)\,d\lambda. \]

The principal role in the proof of Theorem 1 is played by the following

Lemma 2. Let \(a(\lambda)\) be an odd, three times differentiable function with bounded third derivative, vanishing outside \([-1,1]\).

Then, as \(t\to\infty\), uniformly in \(\mu\),

\[ \frac{1}{t}\left|\int_{-\pi}^{\pi} \frac{\sin^2(t\lambda/2)}{\sin^2(\lambda/2)} \,g_{ij}(\lambda+\mu)\,a(t\lambda)\,d\lambda\right| =o\left(\sqrt{h_i(t)h_j(t)}\right). \tag{2} \]

The proof of the lemma is too cumbersome to present here; we note only that it is based on ideas similar to those used in \({}^{1}\) in the proof of Lemma 3.3.

Let now \(\varepsilon>0\) be an arbitrary positive number. Define an even, three-times differentiable function \(a_\varepsilon(\lambda)\) so that on \([\varepsilon,1-\varepsilon]\) \(a_\varepsilon(\lambda)\) coincides with \((\lambda/2)^2\sin^{-2}(\lambda/2)\), is equal to zero outside \([-1,1]\), increases monotonically on \([-\varepsilon,\varepsilon]\), and decreases monotonically on \([1-\varepsilon,1]\). Substituting in (2) \(a_\varepsilon\) in place of \(a\), using the properties of the functions \(h_i(t)\) proved in \({}^{1}\), and arguing in the same way as at the end of the proof of Theorem 1 of \({}^{1}\), we arrive at (1), in any case, if \(i\ne j\). In the case \(i=j\), equality (1) follows from \({}^{1}\).

From Theorem 1 there immediately follows the following result, which is a strengthening of Theorem 4 of \({}^{4}\).

The functions \(f_{ij}(\lambda)\) have no discontinuities of the first kind at those points \(\lambda_0\) where

\[ \int_{\lambda_0-\delta}^{\lambda_0+\delta} f_{ii}(\lambda)\,d\lambda\cdot \int_{\lambda_0-\delta}^{\lambda_0+\delta} f_{jj}(\lambda)\,d\lambda =O(\delta^2),\qquad \delta\to0. \]

Theorem 2 (cf. \({}^{4}\)). Let the spectral density \(\mathbf f(\lambda)\) of the process
\(\boldsymbol\xi(t)=(\xi_1(t),\ldots,\xi_n(t))\) be representable in the form

\[ \mathbf f(\lambda)=\mathbf P(e^{i\lambda})\mathbf g(\lambda)\mathbf P^*(e^{i\lambda}), \]

where \(\mathbf P(z)\) is a polynomial matrix, all elements of the matrix \(\mathbf g(\lambda)\) are bounded and satisfy condition (1), and moreover

\[ \sum_k w_{ij}^2(2^{-k})<\infty,\qquad 0<m\le \det \mathbf g(\lambda),\qquad 0<m\le g_{ii}(\lambda),\quad i=1,\ldots,n. \]

Then the process \(\boldsymbol\xi(t)\) is completely regular.

1. Let now \(\boldsymbol\xi(t)\) be a process with continuous time \(t\), \(-\infty<t<\infty\). We shall still denote the spectral (matrix) density of the process \(\boldsymbol\xi(t)\) by \(\mathbf f(\lambda)\).

Theorem 3. Whatever positive number \(a\) may be, the spectral density \(\mathbf f(\lambda)\) of a completely regular process \(\boldsymbol\xi(t)\) can be written in the form

\[ \mathbf f(\lambda)=\mathbf P_a(\lambda)\mathbf g(\lambda)\mathbf P_a^*(\lambda), \]

where \(\mathbf P(\lambda)\) is a diagonal polynomial matrix, and the transformed \(G_{ij}(\lambda)\) of the functions \(g_{ij}(\lambda)\), \(g_a=\|g_{ij}\|\), satisfy the condition

\[ \sup_{|\lambda|\le a,\ |h|\le\delta} \frac{\left|G_{ij}(\lambda+h)+G_{ij}(\lambda-h)-2G_{ij}(\lambda)\right|} {\left|G_{ii}(\lambda+h)-G_{ii}(\lambda)\right|^{1/2} \left|G_{jj}(\lambda+h)-G_{jj}(\lambda)\right|^{1/2}} =\omega_{ij}^{(a)}(\delta)\to0. \tag{3} \]

Theorem 4 (cf. \({}^{4}\)). Let the spectral density \(\mathbf f(\lambda)\) of the process \(\boldsymbol\xi(t)\) be representable in the form

\[ \mathbf f(\lambda)=\mathbf B(\lambda)\mathbf g(\lambda)\mathbf B^*(\lambda), \]

where all elements \(B_{ij}(\lambda)\) of the matrix \(\mathbf B(\lambda)\) are entire functions of finite degree, square-summable. Suppose further that all elements of the matrix \(\mathbf g\) are bounded, satisfy condition (3) with \(a=\infty\), and moreover

\[ \sum_k \omega_{ij}^2(2^{-k})<\infty,\qquad 0<m\le \det \mathbf g(\lambda),\qquad 0<m\le g_{ii}(\lambda). \]

Then the process \(\boldsymbol\xi(t)\) is completely regular.

Leningrad State University
named after A. A. Zhdanov

Received
14 III 1968

CITED LITERATURE

\({}^{1}\) I. A. Ibragimov, Theory of Probability and Its Applications, 10, 1 (1965).
\({}^{2}\) I. A. Ibragimov, DAN, 161, No. 4 (1965).
\({}^{3}\) I. A. Ibragimov, DAN, 162, No. 5 (1965).
\({}^{4}\) I. A. Ibragimov, DAN, 175, No. 5 (1967).